The spectral geometric mean of Koufany and Lim (2026), already known to encode the eigenvalue geometry of a pair of positive elements in a Euclidean Jordan algebra, sits at a largely unexplored intersection of quantum information theory and post-quantum cryptography. On the cone of positive-definite Hermitian matrices the spectral mean recovers the Fiedler–Pták construction whose spectrum consists of the geometric means of the eigenvalues of the product; the same construction extends verbatim to any simple Euclidean Jordan algebra, including the exceptional Albert algebra
h3(O). This extension examines three interlocking questions: (i) the relation between the spectral geometric mean and quantum relative entropies (Umegaki, Belavkin–Staszewski, and their Jordan-algebraic analogues); (ii) the possible use of Mostow product metrics
dσ as alternative geometries for quantum state spaces and for divergence-based learning; and (iii) the prospect that the exceptional Jordan structure of the Albert algebra, together with its Mostow decompositions, may furnish new hardness assumptions or algebraic primitives for post-quantum cryptography. Concrete open problems and a stratified ontological reading of the exceptional cone are offered. C S Also available on Academia.edu: https://www.academia.edu/attachments/133701404/download_file?s=portfolio
In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers
C and the other one is over the two elements field
F2. Both constructions have been employed in the past fifteen years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semi-simple Lie groups/algebras. To emphasize this aspect one explains on one hand how well-known results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems (
3 qubits,
3 fermions,
3 bosonic qubits...) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized
N-Pauli group.
We show how to formulate physical theory taking as a starting point the set of states (geometric approach). We discuss the relation of this formulation to the conventional approach to classical and quantum mechanics and the theory of complex systems. The equations of motion and the formulas for probabilities of physical quantities are analyzed. A heuristic proof of decoherence in our setting is used to justify the formulas for probabilities. We show that any physical theory can be obtained from classical theory if we restrict the set of observables. This remark can be used to construct models with any prescribed group of symmetries; one can hope that this construction leads to new interesting models that cannot be build in the conventional framework. The geometric approach can be used to formulate quantum theory in terms of Jordan algebras, generalizing the algebraic approach to quantum theory. The scattering theory can be formulated in geometric approach.